Geodesic
A Generalization of the Menshov--Rademacher Theorem
Matematičeskie zametki, Tome 86 (2009) no. 6, pp. 925-937.

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For a sequence $\{X_n\}_{n\ge1}$ of random variables with finite second moment and a sequence $\{b_n\}_{n\ge1}$ of positive constants, new sufficient conditions for the almost sure convergence of $\sum_{n\ge1}X_n/b_n$ are obtained and the strong law of large numbers, which states that $\lim_{n\to\infty}\sum_{k=1}^nX_k/b_n=0$ almost surely, is proved. The results are shown to be optimal in a number of cases. In the theorems, assumptions have the form of conditions on $\rho_n=\sup_k(\mathsf EX_kX_{k+n})^+$, $$r_n=\sup_k\frac{(\mathsf EX_kX_{k+n})^+}{(\mathsf EX_k^2)^{1/2}(\mathsf EX_{k+n}^2)^{1/2}},$$ $\mathsf EX_n^2$, and $b_n$, where $x^+=x\vee0$ and $n\in\mathbb N$.
Keywords: strong law of large numbers, random variable, almost sure convergence, Menshov–Rademacher theorem, Kolmogorov's 0–1 law.
Mots-clés : second moment 
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P. A. Yaskov. A Generalization of the Menshov--Rademacher Theorem. Matematičeskie zametki, Tome 86 (2009) no. 6, pp. 925-937. http://geodesic.mathdoc.fr/item/MZM_2009_86_6_a10/

[1] Sh. Levental, Kh. Salekhi, S. A. Chobanyan, “Obschie maksimalnye neravenstva, svyazannye s usilennym zakonom bolshikh chisel”, Matem. zametki, 81:1 (2007), 98–111 | MR | Zbl

[2] F. Móricz, “The strong laws of large numbers for quasi-stationary sequences”, Z. Wahrsch. Verw. Gebiete, 38:3 (1977), 223–236 | DOI | MR | Zbl

[3] V. F. Gaposhkin, “Kriterii usilennogo zakona bolshikh chisel dlya klassov statsionarnykh v shirokom smysle protsessov i odnorodnykh sluchainykh polei”, TVP, 22:2 (1977), 295–319 | MR | Zbl

[4] V. F. Gaposhkin, “Skhodimost ryadov, svyazannykh so statsionarnymi posledovatelnostyami”, Izv. AN SSSR. Ser. matem., 39:6 (1975), 1366–1392 | MR | Zbl

[5] R. J. Serfling, “On the strong law of large numbers and related results for quasistationary sequences”, TVP, 25:1 (1980), 190–194 | MR | Zbl

[6] V. F. Gaposhkin, “O poryadke rosta summ neortogonalnykh ryadov”, Anal. Math., 6:2 (1980), 105–119 | DOI | MR | Zbl

[7] T.-C. Hu, A. Rosalsky, A. Volodin, “On convergence properties of sums of dependent random variables under second moment and covariance restrictions”, Statist. Probab. Lett., 78:14 (2008), 1999–2005 | DOI | MR | Zbl

[8] S. H. Sung, “Maximal inequalities for dependent random variables and applications”, J. Inequal. Appl., 2008, Article ID 598319, 10 pp | DOI | MR | Zbl

[9] H. Walk, “Almost sure Cesàro and Euler summability of sequences of dependent random variables”, Arch. Math. (Basel), 89:5 (2007), 466–480 | DOI | MR | Zbl

[10] F. Móricz, “SLLN and convergence rates for nearly orthogonal sequences of random variables”, Proc. Amer. Math. Soc., 95:2 (1985), 287–294 | DOI | MR | Zbl

[11] I. Fazekas, O. Klesov, “A general approach to the strong laws of large numbers”, TVP, 45:3 (2000), 568–583 | MR | Zbl

[12] G. Cohen, M. Lin, “Extensions of the Menchoff–Rademacher theorem with applications to ergodic theory”, Israel J. Math., 148:1 (2005), 41–86 | DOI | MR | Zbl

[13] B. C. Kashin, A. A. Saakyan, Ortogonalnye ryady, Izd-vo AFTs, M., 1999 | MR | Zbl

[14] F. Móricz, K. Tandori, “Counterexamples in the theory of orthogonal series”, Acta Math. Hungar., 49:1–2 (1987), 283–290 | DOI | MR | Zbl